ESTIMATING A SMOOTH MONOTONE REGRESSION FUNCTION

The problem of estimating a smooth monotone regression function m will be studied. We will consider the estimator m(SI) consisting of a smoothing step (application of a kernel estimator based on a kernel K) and of a isotonisation step (application of the pool adjacent violator algorithm). The estimator m(SI) will be compared with the estimator m(IS) where these two steps are interchanged. A higher order stochastic expansion of these estimators will be given which show that m(SI) and m(IS) are asymptotically first order equivalent and that m(IS) has a smaller mean squared error than m(SI) if and only if the kernel function of the kernel estimator is not too smooth.